The evaluation of Selberg character sums
Let \(\tau\) be a multiplicative character of maximal order, and \(\varphi\) be the quadratic character of a finite field \(GF(q)\). Let \(E\) run through all monic polynomials \(E=E(x)=x^ n+e_{n-1} x^{n-1}+\dots e_ 0\) of degree \(n\) over \(GF(q)\), and let \(\Delta_ E\) denote the discriminant of \(E\). Write \[ S_ n(a,b,c)=\sum_ E \tau((-1)^{an} E(0)^ a E(1)^ b \Delta^ c_ E)\varphi(\Delta_ E), \] a Selberg character sum. The author conjectured in 1981 that if none of \(a+b+(n-1+j)c\) (\(0\leq j\leq n-1\)) are divisible by \(q-1\), then \[ S_ n(a,b,c)=\prod_{j=0}^{n-1} {{G(a+jc)G(b+jc)G(c+jc)\overline {G}(a+b+(n-1+j)c)} \over {qG(c)}}, \] where \(G(a)=\sum_{\xi\in GF(q)}\tau(\xi)\psi(\xi)\) and \(\psi\) is a non- trivial additive character of \(GF(q)\). In this paper he proves this, and some other similar conjectures.
scientific article; zbMATH DE number 55792